On Friday I set this puzzle……

Which is a better fit (i.e., results in less wasted space), a square peg in a round hole or a round peg in a square hole?

If you have not tried to solve the puzzle, have a go now.  For everyone else the answer is after the break.

So, the answer is that a round peg fits into a square hole better than a square peg in a round hole.  Ten points to the person that can give the clearest explanation of why this is the case!

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. Mat A says:

I based my conclusion on the calculating the results of the following – in the case of the round peg in the hole, the area not taken up by the round peg of the overall square’s area as a percentage was roughly 24%. The area not taken up by a square when placed in the fitting square was roughly 36%, making the round peg in the square hole more economical.

The radius used for the round peg in the square hole was half the length of the side of the square. The radius used for the round hole was half of the diagonal of the square fitting the round hole (corner to corner across).

2. I used maths and a calculator, too. Don’t really have any simple explanation. It’s a hard thing to intuit, to me anyway.

3. Chris Emerson says:

Assuming a square hole with sides of 1 unit, the circle that fits inside it will take pi * 0.5^2 = 0.785 units squared (so 78.5% efficient).

Assuming a round hole of area 1 unit squared, we can work out its radius to be sqrt(1/pi) = 0.564189…

Using this value with Pythagoras to work out the (equal) sides of a right angled triangle with a hypotenuse of twice the radius of the circle, we get 0.798, making the area of the square peg 0.637: 63.7% efficient.

Thus the round peg in a square hole is more efficient for space.

4. M says:

Explaination: The square and the circle share the same centerpoint. To make calculation easy, start with a square with edges of length 2. The surface of this square is 2 x 2 = 4.

The formula to calculate the surface of circle: radius x radius x pi

1) inscribed: the circle touches the edge of the square. So the radius of the circle is 1.

Surface of circle: 1 x 1 x pi = pi.

Ratio = 4 / pi = 1,273… Makes 27,3% loss of area.

2) circumscribed: the circle touches the corners of the square. So the radius of the circle is (using pythagoras):
square root (1 + 1) = sqrt (2).

Surface of circle: sqrt (2) x sqrt (2) x pi = 2 x pi.

Ratio = 2 x pi / 4 = 1,571… Makes 57,1 % loss of area.

That makes that scenario 1 is more efficient.

And what is the answer to the first question about the riots then?

1. Berhard says:

Yeah! What is the answer to riots?
At least i want to know just in case i want to start one myself!!! :-)>

By the, way each peg fits without any wasted space, it ist only a queston of the size of the sledge hammer you use to hammer it in the danm hole….

5. Bill says:

For a round peg to fit in a square hole the diameter of the circle will equal the length of one side of the square. Area of square minus area of circle is your “wasted” space. Approximately 21.5% wasted space.
The square in the circle will need the diameter of the circle to equal length from corner to opposite corner of the square (the hypotenuse of the triangles formed by cutting the square diagonally- use this to get length of the sides of the square). Area of circle minus area of square is “wasted” space. Approximately 36.3% wasted space.
Circle in square wins with less wasted space. It’s been a while since I took geometry, but that’s how I did it.

6. With circle inside square the area of the square is 4r^2 and the circle is pi*r^2. The difference is (4-pi)*r^2 i.e. about 0.86r^2. With square inside circle the area of the square is 2r^2, circle as before, so the difference is (pi-2)*r^2 i.e. about 1.14r^2. So the difference proportionally will always be greater for the square peg in the round hole.

7. This is not an explanation, but a mathematical proof. If you take a square with an area of 100 square units and inscribe a circle into it, the circle has a diameter of 10 and an area of 78.5 square units. Therefore, the wasted space is 21.5 square units or 21.5%.

If you take a circle with an area of 100 square units and inscribe a square into it, the square has sides of 7.98 units and an area of 63.6 square units. The wasted space is 36.4 square units or 36.4%.

Ergo, there is more wasted space when putting a square peg into a round hole.

1. katie k says:

I have a hard time with math, particularly interpreting mathematical explanations without a visual or hands-on component. This was the answer that both made me understand the problem and showed me clearly how to solve it. If there was a prize, I would give it to Mike. 🙂

8. Anonymous says:

Unplugged area for round peg in square hole = r^2(4-pi)
2r * 2r – pi r^2

Unplugged area for square peg in round hole = r^2(pi – 2)
pi r^2 – 2r^2

Therefore round peg in square hole leaves less of a gap.

r = radius of the round peg / hole

9. Raymond says:

Take a square of side x.
a. The radius of the circle inside the square is x/2
b. The radius of the circle outside the square is x/sqrt(2) (using Pythagoras theorem)

c. Area of square is x^2
d. Area of inside circle is (pi.x^2)/4
e. Area of outside circle is (pi.x^2)/2
f. Area of space between Square in circle inside is x^2 – (pi.x^2)/4
g. Area of the space between the circle outside and square is (pi.x^2)/2 – x^2

Now find the ratio of the space to the hole:
f / c
= (x^2 – (pi.x^2)/4)/x^2
now divide top and bottom by x^2
= 1 – pi/4 or about 1/4

g/e
= ((pi.x^2)/2 – x^2) / (pi.x^2)/2
now divide top and bottom by x^2
= (pi/2 – 1)/(pi/2)
now divide top and bottom by pi/2
= 1 – 2/pi or about 1/3

So a round peg in a square hole leave a gap of about a quarter the hole. A Square peg in a round hole leaves a gap about a third of the hole.

10. Same answer as Mat and Chris but without a calculator:

1. Round peg of radius 1 has area pi; this fits a square hole of width 2, which has area 4; so the amount of space used is pi/4.

Square peg of width 2 has area 4; this fits a round peg of radius sqrt(2) (draw a picture!), which has area 2pi; so the amount of space used is 4/2pi, or 2/pi.

Rounding pi to 3, as they like to do in Indiana, the round peg takes up 3/4 of the space, which is more than the square peg, which takes up 2/3.

11. lehcyfer says:

When you insert wrong peg you get four triangles of empty space in the corners.

In case of square hole/round peg these triangles have two straight walls and one concave – they are “thin”. In case of round hole/square peg these triangles have one straight wall and two bulging ones – they are “fat”.

It’s no wonder that thin triangles take less space than fat ones 😛

12. Michael Sternberg says:

Sure, there are fine and exact mathematical answers, but these are to a rather academic problem involving the incircle or circumcircle of a square.

If such an academic peg fits tightly and merely leaves some corners or circular segments open, then no ink would have been spilled or phrase coined about that. The practical and much more interesting case is when the peg is too large to fit as is. To make it fit, the peg has to be shaved down or the hole has to be widened, or both. That’s six cases to consider (three for each peg type).

The optimal solution for each case will depend on the degree of (a) initial and (b) final tolerable misfit (percent oversize), (c) the hardness and elasticity of the two materials, and (d) the available tools. Let’s leave that for the ages and engineers to figure.

Richard’s clipart image illustrates one of the cases rather nicely, along with an inventive and appropriate solution, and one that epitomizes the connotation of ingenuity expressed by the idiom.

1. Greg23 says:

I knew a Michael Sternberg once. Purdue ’72?

13. Nigel says:

I just guessed circle based on a circle enclosing the greatest area for the least circumference.

14. Karthik Durvasula says:

1) Imagine a small-square within a big-square, such that the small-square touches the mid-points of the sides of the big-square.

3) The area of the big-square is twice that of the small square. (Imagine folding the edges of the big-square onto the small-square. It should fit perfectly. So, the edges together have the area of the small square. If you add the area occupied by the small square to that of the edges, the big-square is twice the small square.)

4) Now, imagine a circle thru the points of contact between the two squares. (it would result in a small-square within a circle, which is within a big-square).

5) If the side of the big-square = 2r;
area of the big-square is 4r^2;
area the small-square = half(big-square) = 4r^2/2 = 2r^2
area of the circle = pi*r^2

6) 2r^2 > pi*r^2 > 4r^2
Area(small-square) > Area(circle) > Area(big-square)

7) pi is closer to 4 than to 2. Therefore, the area of the circle is closer to that of the big-square than the small-square.

8) Therefore, a circle in a square will result in less wastage than a square in a circle.

9) Meaning: round-peg in a square hole wastes less than a square-peg in a round hole.

15. Karthik Durvasula says:

Correction:

1) Imagine a small-square within a big-square, such that the small-square touches the mid-points of the sides of the big-square.

2) The area of the big-square is twice that of the small square. (Imagine folding the edges of the big-square onto the small-square. It should fit perfectly. So, the edges together have the area of the small square. If you add the area occupied by the small square to that of the edges, the big-square is twice the small square.)

3) Now, imagine a circle thru the points of contact between the two squares. (it would result in a small-square within a circle, which is within a big-square).

4) If the side of the big-square = 2r;
area of the big-square is 4r^2;
area the small-square = half(big-square) = 4r^2/2 = 2r^2
area of the circle = pi*r^2

5) 2r^2 > pi*r^2 > 4r^2
Area(small-square) > Area(circle) > Area(big-square)

6) pi is closer to 4 than to 2. Therefore, the area of the circle is closer to that of the big-square than the small-square.

7) Therefore, a circle in a square will result in less wastage than a square in a circle.

8 ) Meaning: round-peg in a square hole wastes less than a square-peg in a round hole.

1. M says:

Nice!

16. Charles Sullivan says:

Cover bottom of round hole, then fill hole with liquid. Squeeze square peg into round hole, collecting the liquid that squirts out. Measure volume of liquid that squirted out in measuring cup (or beaker).

Now, do the same with square hole and round peg.

Whichever of these two procedures produces the most liquid has the least amount of wasted space.

1. joshthelegend says:

good theory!

17. Dharmaruci says:

You can use some hypertheticals to get there….

Imagine you wanted to fit the best rectangle into a circle. That would clearly be a square, right?

But the best circle into a rectangle would be an ellipse, yes?

So squares are better / more general that circles as they fit the general case better.

18. vitamentis says:

I just filled the cracks with water and measured which had less.

19. Duckfish says:

square in circle ratio
Square: circle = 2 : 3.14(pi) = 63.66%
Reason: assume radius is 1, circle area is 1 pi
Then 45 45 90 triangle side ratio is 1;1; root2 so square side length is
Root 2, so square area is 2. ( 45 45 90 triangle is 1/4 square)

Circle in square ratio
Circle : square = 3.14(pi) : 4 = 0.785
reason: assume radius is 1, circle area is 1 pi
Side length is diameter is 2 area is 4

20. Duck fish says:

78.5% might look better than 0.785….I forgot I use percentage at the previous one

21. ivan says:

I rather suspected the riot question was a joke, but I also hoped it was going to have was a mathermatical answer. If you don’t believe in free-will, as many scientists don’t, then human behaviour no more has a “cause” than the motion of rocks around the sun, so events are contingent only on the outcome of those quantum events which are random.

22. SeleneBlueSky7 says:

“So, the answer is that a round peg fits into a square hole better than a square peg in a round hole.”
Of course. I just think square pegs are usually bigger in diameter than round pegs.
No need for calculations. Usually round pegs go to square holes better. 🙂

23. CAPER says:

As the circle of same “width and height” lacks the corners of the square, it is the better fit. You have to cut away the squares’ corners to reveal the circle within..!!!!

24. Mike says:

Consider a square in a diamond in a square. Clearly the square fits in the diamond identically to the diamond in the square. And padding out the diamond into a circle makes it an even better fit and the square into the circle an even worse one. So round peg in square hole wins.

1. Anonymous says:

I like this! Easy way to think about it, without requiring calculations. It never occurred to me. And here I did it the long way….

25. Jake says:

Yes, Knew It.
Very simple explanation.

For both cases, radius of the peg=1, therefore, area of the circle equals pi:
r=1 + A=(pi)r^2 ——> A=(pi)(1)^2=PI=3.14

Now, case #1, circle in square. The radius of the circle is half the length of a square, therefore, a side of a square=2, thus the area of square is 4
s=2 + A=s*s ———> A=2*2=4

Now, case #2, square in circle. Since the radius is 1, this means that the diameter of the circle is 2, which is also a diagonal of the square. Therefore the area of this square is 2.
d=2 + A=.5*d*d ————-> A=.5*2*2=2

Now, to see what it means.
In case 1, the area of the circle in the square is (pi)/4 which is roughly 78.5%
In case 2, the area of the square in the circle is 2/(pi) which is roughly 63.7%

THEREFORE, since the circle in the square takes up more space, it would
be a better fit.

🙂

26. ChivNick says:

Proportion of wastage space with circle c in square s is (s-c)/s

Proportion of wastage space with round in square is (c-s)/c

Note – we are dealing in proportions here, so the actual sizes don’t matter.

In terms of the radius r of a circle, the area of the circle is always 3.14r^ (peg or hole), and the square is either (2r)^ (round in square) or 2r^ (square in round).

Solve for an easy value of r (I chose 2) then plug the answer into the proportion calculation at the top. The bigger value is the most wasteful.

Q.E.D.

1. ChivNick says:

I should have said that the answer is:

21% waste circle in square

36% waste square in circle

27. PirateDuzzo says:

Check this clip out at about 30 seconds in.

A round peg fitting perfectly in a square hole. No wasted space.

28. Ian says:

Richard, your pop up box asking me to subscribe to your blog is completely blocking my view of your posts. I’m on a mobile phone and cannot close it. Love this site, but that damn box makes this site impossible to use!!! Please get rid of the godamn thing.

29. Mark_D says:

I took the ratios of the areas of the circles and squares (Area of Peg divided by Area of Hole). That ratio is the %Efficiency. I kept everything in terms of the circle’s diameter.

As = Area Square; Ac = Area Circle; D = diameter

Square in Circle: As = (D^2)/2; Ac = (D^2 * pi) / 4 –> As/Ac = 2/pi = 0.64, or 64% Efficient

Circle in Square: As = D^2; Ac = (D^2 * pi) / 4 –> Ac/As = pi/4 = 0.79, or 79% Efficient

Therefore, Circle in Square is ~15% more efficient than Square in Circle.

30. Norman says:

Now just slice both cases in 4 pieces (like +).
In your circle into square case you have 4 holes.
In your square into cirlce you have 8 holes.

Maybe pretty non-mathematical but easy to visualize 🙂

31. rmb says:

Using simple geometric formulas, it can be deduced that the square peg in round hole would cover 64% of the hole area and the round peg in square hole would cover 79% of the hole area. So round peg in square hole is a better fit.

32. Cause of the riots;

“Social proof from similar others. ” Cialdini.

Sounds about right to me. He says it’s not about what YOU think is right or wrong, it’s about what you think OTHER people believe.

That’s why people smoke who know its bad for them, it’s why people do all kinds of non-intuitive things. Including religion. Clever stuff

33. Great puzzle. My simple answer is because the diameter of a circle that fits inside a square is equal to the square’s length. The diameter of the circle that fits outside of a square is that square’s diagonal, (which is always sqrt(2) times the length of the square).

The percentage of wasted space is the (area of the hole minus the area of the peg) divided by the area of the hole.

For a round peg, that’s (d^2 – d^2*pi/4)/d^2 = (1- pi/4) or ~21%

For a square peg, that’s (D^2*pi/4 – (D/sqrt(2))^2) / (D^2*pi/4)
= 4*(D^2*pi/4 – D^2/2)/D^2*pi
= 1 – 2/pi
~36%

34. You don’t have to use any maths. Just draw a circle in a square and a square in a circle. You can see more space wasted with a square in a circle with your naked eye.

1. Anonymous says:

I agree no maths needed, but there is no need for drawing either. Imagine a circle and a square of the same maximum diameter, therefore the same radius. It is a property of circles that they enclose the most surface area of any shape for a given radius. Simple.

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